### Fast imaging on the erasure detection subspace

Here we explain how we carry out the erasure imaging that permits us to discover site-resolved leak mistakes^{30} To both prevent any additional heating originating from the imaging beams and enhance the imaging fidelity, we shine 2 similar counter-propagating beams with crossed π-polarization and Rabi frequencies of * Ω*/ 2* π* ≈ 40 MHz on the ^{ 1} S_{ 0} →^{ 1} P_{ 1} shift (Extended Data Fig. 1a). This reduces the net force on an atom, and the crossed polarization prevents strength disturbance patterns.

We highlight the particular functions of this imaging plan experimentally. We display in Extended Data Fig. 1b the survival possibility of atoms in ^{ 1} S_{ 0} as a function of imaging time. After 4 μs, more than 80% of the atoms are lost. The number of identified photons continues to increase: even though the kinetic energy of the atoms is too big to keep them caught, their mean position stays centred on the tweezers. Significantly, for our execution of erasure excision, atom loss throughout the erasure image is irrelevant for our functions as long as the preliminary existence of the atom is properly recognized, however in any case, other quick imaging plans might reduce this result^{51} After about 24 μs, the atomic spread ends up being too big and the variety of identified photons plateaus. The acquired detection pie chart is displayed in Extended Data Fig.1c We provide the outcomes both for empty (blue) and filled (red) tweezers, which we attain by very first imaging the atoms utilizing normal, high survival imaging for preliminary detection in a 50% packed range, then carry out the quick image. We acquire a common detection fidelity of ( 0.98 {0} _ {-1} ^ {+1} ) of real negatives and real positives, restricted by the limited possibility for atoms in ^{ 1} P_{ 1} to decay into ^{ 1} D_{ 2} (Extended Data Fig. 1a).

This imaging plan is adequately quick to prevent troubling atoms in ^{ 3} P_{ 0}, as determined by losses from ^{ 3} P_{ 0} as a function of imaging time (Extended Data Fig. 1d). We fit the information (circles) utilizing a direct function (strong line), and acquire a loss of ( 0.000004 {6} _ {-12} ^ {+12} ) per image, constant with the life time of the ^{ 3} P_{ 0} state^{52} of about 5 s for the trap depth of 45 μK utilized throughout quick imaging.

As to the nature of the identified erasure mistakes for the Bell state generation, we discover that preparation mistakes contribute the huge bulk of erasure occasions compared to brilliant Rydberg decay, and excising them has a more considerable influence on minimizing extramarital relations. In specific, application of ( widehat {U} ) lasts for just about 59 ns, which is substantially much shorter than the individually determined brilliant state decay life time of ( 16 {8} _ {-14} ^ {+14}, {rm {mu}} {rm {s}} ) (Extended Data Fig. 2). The mistake design explained in Fig. 2 recommends that excising such mistakes leads to an adultery decrease of just ( 1. {2} _ {-3} ^ {+3} times 1 {0} ^ {-4} ) (Methods). Alternatively, preparation mistakes represent about 5 × 10^{ − 2} adultery per set due to the very long time in between preparation in ( left|grightrangle ) and Rydberg excitation (Extended Data Fig. 3). The gains in fidelity from erasure conversion generally come from getting rid of almost all the preparation mistakes, which has actually the included advantage of substantially minimizing mistake bars on the SPAM-corrected worths. Still, SPAM-corrected worths may likewise take advantage of the little gain in getting rid of the result of brilliant state decay, and from preventing prospective unhealthy impacts occurring from greater atomic temperature level in the repumper case.

For erasure detection utilized in the context of many-body quantum simulation, we change the binarization limit for atom detection to raise the false-positive imaging fidelity to 0.9975, while the false-negative imaging fidelity is decreased to about 0.6 (Fig. 3d); this is done as a conservative procedure to focus on making the most of the variety of functional shots while possibly passing up some fidelity gains (Extended Data Fig. 7).

We keep in mind that the plan we reveal here is not yet basically restricted, and there are a variety of technical enhancements that might be made. The cam we utilize (Andor iXon Ultra 888) has a quantum effectiveness of about 80%, which has actually been enhanced in some current designs, such as quantitative complementary metal oxide semiconductor (qCMOS) gadgets. Even more, we presently image atoms from just one instructions, when, in concept, photons might be gathered from both goals^{53} This would enhance our approximated overall collection effectiveness of about 4% by an element of 2, causing quicker imaging times with greater fidelity (as more photons might be gathered before that atoms were ejected from the trap). The fidelity might be significantly enhanced by actively repumping the

### 1

D^{13} 2* state back into the imaging manifold to not efficiently lose any atoms by means of this path.*^{ Details of Rydberg excitation} Our Rydberg excitation plan has actually been explained in depth formerly^{ Before the Rydberg excitation, atoms are initialized from the outright ground state 5} s_{ 2}* 1* S* 0* to the metastable state 5^{ s} 5_{ p}

3* P* 0* (698.4 nm) through meaningful drive. Consequently, tweezer trap depths are decreased by an element of 10 to extend the metastable state life time.* For Rydberg excitation and detection, we snuff out the traps, drive to the Rydberg state (5^{ s} 61_{ s}* 3*_{ S} 1*,*_{ m} J^{13}= 0, 31 nm), where ^{ m} J_{ is the magnetic quantum variety of the overall angular momentum, and lastly carry out auto-ionization of the Rydberg atoms} Auto-ionization has a particular timescale of about 5 ns, however we carry out the operation for 500 ns to guarantee overall ionization. We report a more precise measurement of the auto-ionization wavelength as about 407.89 nm. In the last detection action, atoms in ^{13 354} P

0^{ read out by means of our typical imaging plan},_{} Atoms can decay from ^{13} 3^{ P} 0_{ in between state preparation and Rydberg excitation, which is 60 ms to permit time for the electromagnetic fields to settle. In previous work}, we supplemented meaningful preparation with incoherent pumping to

### 3

P

0

right away before Rydberg operations. Throughout the repumping procedure, atoms can be lost due to duplicated recoil occasions at low trap depth, which is not identified by the erasure image, and therefore can decrease the bare fidelity. Even with SPAM correction of this result, we anticipate the fidelity with repumping to be somewhat inferior owing to an increased atomic temperature level for pumped atoms.

Rydberg Hamiltonian* The Hamiltonian explaining a selection of Rydberg atoms is well estimated by*$$ hat {H}/ hslash =frac {varOmega} {2} amount _ {i} {hat {X}} _ {i} -varDelta amount _ {i} {hat {n}} _ {i} +frac {{C} _ {6}} {{} ^ {6}} amount _ {i > > j} frac {{hat {n}} _ {i} {hat {n}} _ {j}} {| i-j ^ {6}} $$*
( 2 ).
* which explains a set of engaging two-level systems, identified by website indices

*i*and

*j*, driven by a laser with Rabi frequency

*Ω*

_{ and detuning }Δ

*The interaction strength is figured out by the*C 6 coefficient and the lattice spacing a Operators are ( {widehat {X}} _ {i} = { rrightrangle} _ {i} _ {i} + { grightrangle} _ {i} _ {i} ) and ( {widehat {n}} _ {i} = { rrightrangle} _ {i} _ {i} ), where

*( { grightrangle} _ {i} )*and

*( { rrightrangle} _ {i} )*signify the metastable ground and Rydberg states at website

i*, respectively, and * ℏ2a is the decreased Planck continuous.* For the case of determining two-qubit Bell state fidelities, we set *^{ Ω}/ 2π= 6.2 MHz. Interaction strengths in Fig. ^{55} are straight determined at interatomic separations of 4 μm and 5 μm, and theorized by means of the forecasted 1/* r*_{ 6} scaling to the level at 2.5 μm. Mean atomic ranges are adjusted by means of a laser-derived ruler based upon moving atoms in meaningful superposition states^{ We adjust } C^{18} 6

/ 2π= 230( 25 )GHz μm* 6* utilizing optimum probability estimate (and associated unpredictability) from resonant quench characteristics, which in addition adjusts an organized balanced out in our international detuning. For carrying out many-body quasi-adiabatic sweeps, the detuning is swept symmetrically in a tangent profile from +30 MHz to − 30 MHz, while the Rabi frequency is efficiently switched on and off with an optimum worth of Ω/ 2π= 5.6 MHz. For an at first favorable detuning, the ( left|rrightrangle ) state is energetically beneficial, making the all-ground preliminary state, ( left|gg … ggrightrangle ), the greatest energy eigenstate of the blockaded energy sector, where no neighbouring Rydberg excitations are permitted. For unfavorable detunings, where ^{39}( left|grightrangle )^{56} is energetically beneficial, the greatest energy state distinctively ends up being the symmetric AFM state

### (( left|grgr … grrightrangle +left|rgrg … rgrightrangle )/ sqrt {2} )

in the deeply bought limitation. Therefore, thinking about just the blockaded energy sector, sweeping the detuning from favorable to unfavorable detuning (therefore staying in the greatest energy eigenstate) is comparable to the ground-state physics of an efficient Hamiltonian with appealing Rydberg interaction and inverted indication of the detuning. This equivalence permits us to run in the efficiently appealing program of the blockaded stage diagram of ref.^{13 For our Hamiltonian specifications, we utilize specific diagonalization numerics to determine the infinite-size important detuning utilizing a scaling collapse near the finite-system size minimum energy space18}^{57} Error modelling2 Our mistake design has actually been explained formerly^{18},

### We carry out Monte Carlo wavefunction-based simulations

, accounting for a range of sound sources consisting of time-dependent laser strength sound, time-dependent laser frequency sound, tasting of the beam strength from the atomic thermal spread, Doppler sound, variations of the interaction strength from thermal spread, beam pointing stability and others. All of the specifications that get in the mistake design are individually adjusted by means of selective measurements straight on an atomic signal if possible, as displayed in Extended Data Table^{13} Criteria are not fine-tuned to match the determined Bell state fidelity, and the design similarly well explains arise from many-body quench experiments_{ Extraction of the Bell state fidelity To draw out the Bell state fidelities priced estimate in the primary text, we utilize a lower-bound approach}, which depends on determining the populations in the 4 possible states * P*_{ g r},* P*_{ r g},* P*_{ g g} and P r r throughout a Rabi oscillation in between

( left|ggrightrangle )

and

( left|{varPsi} ^ {+} rightrangle ) The lower bound on Bell state fidelity is provided by: $$ {F} _ {{rm {Bell}}} ge frac {{P} _ {gr+ rg} ^ {{rm {pi}}}} {2} +sqrt {frac {{amount} _ {i} {{left( {|( {} P} _ {i} ^ {2 {rm {pi}}} {right)} |)}} ^ {2} -1} {2} + {P} _ {gr} ^ {{rm {pi}}} {P} _ {rg} ^ {{rm {pi}}}},$$

( 3 ).

where *( {P} _ {i} ^ {2 {rm {pi}}} )*_{ are the determined likelihoods for the 4 states at 2π, and ( {P} _ {gr+ rg} ^ {{rm {pi}}} )} is the possibility * P*_{ g r}+5a P r g* determined at π. To determine these likelihoods with high precision, we focus our data-taking around the π and 2π times (Extended Data Fig. *), and fit the acquired worths utilizing quadratic functions *( f( t)= {p} _ {0} + {p} _ {1} {(t- {p} _ {2})} ^ {2} )*_{, where } t* is time, and (*_{ p} 0*, *_{ p} 1

#### ,

p* 2*) are totally free specifications. We initially information the fitting approach, then how we acquire the 4 likelihoods, and lastly the extraction of the Bell state fidelity from these.* Fitting approach*_{ We carry out a fit that considers the underlying beta circulation of the information and avoids methodical mistakes occurring from presuming a Gaussian circulation of the information. The goal of the fit is to acquire the three-dimensional possibility density function } Q*(*_{ p} 0*,*_{ p} 1*,* p* 2*) of f, utilizing each speculative information point * i* specified by its possibility density function ( {{mathcal {P}}} _ {i} (x)), where x is a possibility. To acquire a specific worth of *( Q( {widetilde {p}} _ {0}, {widetilde {p}} _ {1}, {widetilde {p}} _ {2} ))*, we take a look at the matching possibility density function worth ( {{mathcal {P}}} _ {i} (, f( {t} _ {i} ))) for each information point i, where

( f( {t} _ {i} )= {widetilde {p}} _ {0} + {widetilde {p}} _ {1} {({t} _ {i} – {widetilde {p}} _ {2})} ^ {2} )

, and appoint the item of each

( {{mathcal {P}}} _ {i} (, f( {t} _ {i} ))) to the fit probability function: [{widetilde{p}}_{0},{widetilde{p}}_{1},{widetilde{p}}_{2}]$$ Q( {widetilde {p}} _ {0}, {widetilde {p}} _ {1}, {widetilde {p}} _ {2} )= prod _ {i} {{mathcal {P}}} _ {i} (, f( {t} _ {i} )).$$

( 4 ).

We duplicate this for different 5b()[*p*_{0}, *p*_{1}, *p*_{2}] The outcome of such fitting approach is displayed in Extended Data Fig. * (black line), where we provide *( f( t)= {p} _ {0} + {p} _ {1} {(t- {p} _ {2})} ^ {2} )* for *_{ representing the optimum worth of } Q*(*_{ p} 0*,*_{ p} 1

#### ,

p* 2*_{). We stress that this leads to a lower peak worth than a basic fitting treatment that presumes underlying Gaussian circulations of experimentally determined likelihoods (red line). Selecting this lower peak worth ultimately will offer a more conservative however more precise worth for the Bell state fidelity lower bound than the ignorant Gaussian method. Obtaining the 4 possibility circulations} Our approach to acquire the possibility density functions of the 4 likelihoods at π and 2π times makes sure both that the amount of the 4 likelihoods constantly equates to one which their shared connections are maintained. We initially draw out the beta circulation of 5c P* r*_{ r by collecting all the information around the π and 2π times (Extended Data Fig. }). In specific, the mode of the acquired beta circulation at π is * P*_{ r r ≈ 0.0005. The circulation of P g} r*+*_{ r g} and * P*_{ g g are acquired by fitting the information in the list below method. We carry out a joint fit on P g} r*+*_{ r} g* utilizing an in shape function * f* 1*_{( t}), and on * P*_{ g} g* utilizing an in shape function * f

2

(

t

). The healthy functions are revealed as:

$$ {f} _ {1} (t)= {p} _ {0} + {p} _ {1} {(t- {p} _ {2})} ^ {2},$$

( 5 ).

_{$$ {f} _ {2} (t)= 1- {p} _ {0} – {P} _ {rr} – {p} _ {1} {(t- {p} _ {2})} ^ {2},$$}

( 6 ).

* which makes sure that the amount of the 4 likelihoods is constantly equivalent to 1. We then compute the joint possibility density function *_{ Q} 1,2*(*_{ p} 0*,*_{ p} 1*,*_{ p} 2*) of both *_{ f} 1

and

f

2 utilizing the approach explained above. In specific: $$ {Q} _ {1,2} ({widetilde {p}} _ {0}, {widetilde {p}} _ {1}, {widetilde {p}} _ {2} )= prod _ {i} {{mathcal {P}}} _ {i} ^ {gr+ rg} ({f} _ {1} ({t} _ {i} )) prod _ {i} {{mathcal {P}}} _ {i} ^ {gg} ({f} _ {2} ({t} _ {i} )),$$

( 7 ).

where *( {{mathcal {P}}} _ {i} ^ {gr+ rg} )*_{ (( {{mathcal {P}}} _ {i} ^ {gg} )) is the possibility density function related to P g} r*+*_{ r g} (* P* g* g*_{) for the } i* th speculative information point. In specific, we enforce that *_{ p 0} ≤ 1 −* P*_{ r} r* to prevent unfavorable likelihoods. We reveal the resulting *_{ Q} 1,2*(*_{ p} 0*,*_{ p} 15d,* p*_{ 2}) in Extended Data Fig. * as two-dimensional maps along (*_{ p} 0*,*_{ p} 1*) and (*_{ p} 0

,* p*_{ 2}).* We then acquire the one-dimensional possibility density function for *_{ p} 0* by incorporating over *_{ p} 15d and * p*_{ 2 (Extended Data Fig. ). This offers the fitted possibility density function of P g} r*+*_{ r g}, and for this reason * P*_{ g g}= 1 −* P*_{ r r} −* P*_{ g r} −* P*_{ r g} at π time. We duplicate this procedure for different worths of

P* r*_{ r, for both π and 2π times.} At the end of this procedure, we acquire various possibility density functions for each * P*_{ r r} worth. The asymmetry in between * P*_{ g r} and * P*_{ r g} is acquired by taking the mean of * P*_{ g r} −* P*_{ r g} at π and 2π times. We presume the hidden circulation to be Gaussian, as * P*_{ g r} −

#### P

r* g*_{ is centred on 0, and can be unfavorable or favorable with equivalent possibility. Bell state fidelity} Now that we have the possibility density function for all 4 likelihoods at π and 2π times, we carry on to the Bell state fidelity extraction. For both π and 2π, we carry out a Monte Carlo tasting of the beta circulation of * P*_{ r r, which then results in a joint possibility density function for P g} r*+*_{ r g} and 3 P5e g g We then sample from this, and utilize formula (

) to acquire a worth for the Bell state fidelity lower bound. We duplicate this procedure 1 million times, and fit the acquired outcomes utilizing a beta circulation (Extended Data Fig. ^{13}). We observe an outstanding contract in between the fit and the information, from which we acquire 2( {F} _ {{rm {Bell}}} ge 0.996 {2} _ {-13} ^ {+10} )5f, where the priced estimate worth is the mode of the mistake and the circulation bars represent the 68% self-confidence period. We utilize the very same approach to acquire the measurement-corrected Bell fidelity and the SPAM-corrected one. After drawing the likelihoods from the possibility density functions, we presume the SPAM-corrected likelihoods from our recognized mistakes, explained in information formerly We utilize the worths reported in Extended Data Table

### Throughout this procedure, there is a limited possibility that the amount of likelihoods does not summarize to one. This originates from the reality that the possibility density functions and the SPAM correction are uncorrelated, a problem that is prevented for raw Bell fidelity extraction owing to the associated fit treatment explained above. We utilize a kind of rejection tasting to reduce this concern by rebooting the entire procedure when it comes to such occasion. We perform this 1 million times, and fit the acquired outcomes utilizing a beta circulation (Extended Data Fig.

). We observe an outstanding contract in between the fit and the information, from which we acquire a SPAM-corrected fidelity

( {F} _ {{rm {Bell}}} ge 0.998 {5} _ {-12} ^ {+7} )

, where the priced estimate worth is the mode of the mistake and the circulation bars represent the 68% self-confidence period.

Interaction restriction for Bell fidelity* We approximate the in theory anticipated Bell state fidelity utilizing perturbation analysis. Particularly, the resonant blockaded Rabi oscillation for an engaging atom set is explained by the following Hamiltonian*$$ widehat {H}/ hbar =frac {varOmega} {2} ({widehat {X}} _ {1} + {widehat {X}} _ {2} )+ V {widehat {n}} _ {1} {widehat {n}} _ {2},$$

( 8 ).

_{ where } V*=*^{ C} 6*/* r2 6 is the distance-dependent, interaction strength in between 2 atoms separated at range r (formula ()). As the two-atom preliminary ground state, ( left|psi (0 )rightrangle =left|ggrightrangle ), has even parity under the left– best reflection proportion, the Rabi oscillation characteristics can be efficiently fixed in an even-parity subspace with 3 basis states of ( left|ggrightrangle ), *( left|rrrightrangle )* and ( left|{varPsi} ^ {+} rightrangle =frac {1} {sqrt {2}} (left|grrightrangle +left|rgrightrangle )) In the Rydberg-blockaded program where * V* ≫

Ω

, we can carry out perturbation analysis with the perturbation specification *( eta =varOmega/ sqrt {2} V)*_{ and discover that the energy eigenvectors of the subspace are estimated as}$$ start {range} {l} {E} _ {1} rightrangle approx frac {{left( 1-frac {|( 1-frac {} eta} {4} -frac {{eta} ^ {2}} {32} ) left ggrightrangle +left( -1- frac {eta} {4} +frac {17 {eta} ^ {2}} {32} ) left {varPsi} ^ {+} rightrangle +left( eta -frac {3 {eta} ^ {2}} {4} ) left rrrightrangle} {sqrt {2}} left|{E} _ {2} rightrangle approx frac {{left( -1- frac {|( -1- frac {} eta} {4} +frac {{eta} ^ {2}} {32} ) left ggrightrangle +left( -1+ frac {eta} {4} +frac {17 {eta} ^ {2}} {32} ) left {varPsi} ^ {+} rightrangle +left( eta +frac {3 {eta} ^ {2}} {4} ) left rrrightrangle} {sqrt {2}} left|{E} _ {3} rightrangle approx {eta} ^ {2} ggrightrangle +eta left|{varPsi} ^ {+} rightrangle +left|rrrightrangle end {range} $$* with their matching energy eigenvalues of * E* 1* ≈* V*^{( −} η* −*_{ η} 2*/ 2), * E* 2* ≈* V*^{(} η* −*_{ η} 2*/ 2) and * E* 3*^{ ≈} V

( 1+

η

2*/ 2), respectively. Rewording the preliminary state utilizing the annoyed eigenbasis, we resolve*_{$$ {F} _ {{rm {Bell}}} =mathop {max} limitations _ {t}|langle {varPsi} ^ {+}|{{rm {e}}} ^ {- {rm {i}} widehat {H} t}|psi (0 )rangle ^ {2} $$}

( 9 ).

* to acquire the analytical expression of the optimum attainable Bell state fidelity, * F* Bell*, at a provided perturbation strength

η

Keeping the option approximately the 2nd order of

η, we discover$$ {F} _ {{rm {Bell}}} =1-frac {5} {4} {eta} ^ {2} =1-frac {5} {8} {{left( frac {|( frac {} varOmega} {V} {right)} |)}} ^ {2} $$

###
( 10 ).

Statistics decrease due to erasure excision(*) Our presentation of erasure excision clearly disposes of some speculative awareness (Extended Data Fig. (*)), which can be viewed as a disadvantage of the approach. This is a manageable compromise: by changing the limit for finding an erasure mistake, we can stabilize gains in fidelity versus losses in speculative data (as revealed in Extended Data Fig. (*)) for whatever specific job is of interest. In basic, the maximum most likely constantly consists of some quantity of erasure excision, as it is typically much better to get rid of incorrect information than keeping them.(*)